Rearranging the Alternating Harmonic Series

By Larry Riddle, Agnes Scott College

Why are conditionally convergent series interesting? While mathematicians
might undoubtably give many answers to such a question, Riemann's theorem on
rearrangements of conditionally convergent series would probably rank near the
top of most responses. Conditionally convergent series are those series that
converge as written, but do not converge when each of their terms is replaced
by the corresponding absolute value. The nineteenth-century mathematician Georg
Friedrich Bernhard Riemann (1826-1866) proved that such series could be
rearranged to converge to any prescribed sum. Almost every calculus text
contains a chapter on infinite series that distinguishes between absolutely and
conditionally convergent series. Students see the usefulness of studying
absolutely convergent series since most convergence tests are for positive
series, but to them conditionally convergent series seem to exist simply to
provide good test questions for the instructor. This is unfortunate since the
proof of Riemann's theorem is a model of clever simplicity that produces an
exact algorithm. It is clear, however, that even with such a simple example as
the alternating harmonic series one cannot hope for a closed form solution to
the problem of rearranging it to sum to an arbitrary real number. Nevertheless,
it is an old result that for any real number of the form ln(sqrt(r)),
where r is a rational number, there is a very simple rearrangement of
the alternating harmonic series having this sum. Moreover, this rearrangement
shares, at least in the long run, a nice feature of Riemann's rearrangement,
namely that the partial sums at the breaks between blocks of different sign
oscillate around the sum of the series.

If we rearrange the alternating harmonic series so that instead of alternating
1 positive odd reciprocal with 1 even reciprocal, we alternate blocks of 375
consecutive odd reciprocals (the positive terms) with blocks of 203 consecutive
even reciprocals (the negative terms), we will get a series that converges to 1
(well, almost) as depicted in Figure 1 (the vertical axis is not to scale in
this figure). The turns in Figure 1 indicate the partial sums at the break
between the positive and negative blocks. The partial sums seem to oscillate
around 1. Now if we rearrange the series with blocks of 1 positive term
followed by blocks of 36 negative terms, we will get a series that converges
exactly to -ln(3). However, as we see from Figure 2, for this
rearrangement the partial sums at the breaks require some initial posturing
before they appear to start oscillating around the sum.

Figure 1: p = 375, n = 203 and Figure 2: p = 1, n =
36

How did we know to choose 375 and 203 in the first example, or 1 and 36 in the
second? Why does the first example almost converge to 1, but the second
converges exactly to -ln(3)?
Why do these partial sums converge in the manner depicted in Figures 2 and 3?
The purpose of this article is to address these issues by considering the
following questions:

What is the sum of a "simple" rearrangement of the alternating harmonic
series?

How should the series be rearranged to obtain a sum within a given
tolerance of a specified value?

What is the behavior of the convergence of these simple
rearrangements?

The latter two questions are completely answered by Riemann's theorem for
rearrangements of arbitrary conditionally convergent series, but our goal is to
provide a more concrete setting of Riemann's results within the context of the
alternating harmonic series with the hope that the reader will then have a
better understanding of the general theory. As a bonus, we will encounter
along the way some interesting mathematics in the form of Riemann sums, Euler's
constant, continued fractions, geometric sums, graphing via derivatives, and
even Descarte's Rule of Signs.